scholarly journals Symbolic Computing in Probabilistic and Stochastic Analysis

2015 ◽  
Vol 25 (4) ◽  
pp. 961-973 ◽  
Author(s):  
Marcin Kamiński
Keyword(s):  
Stochastic Analysis ◽  
Automatic Differentiation ◽  
Perturbation Technique ◽  
Finite Difference Methods ◽  
Stochastic Perturbation ◽  
Structural Response ◽  
Symbolic Computing ◽  
Response Function Method ◽  
Stochastic Perturbation Technique ◽  
Analytical Approaches

Abstract The main aim is to present recent developments in applications of symbolic computing in probabilistic and stochastic analysis, and this is done using the example of the well-known MAPLE system. The key theoretical methods discussed are (i) analytical derivations, (ii) the classical Monte-Carlo simulation approach, (iii) the stochastic perturbation technique, as well as (iv) some semi-analytical approaches. It is demonstrated in particular how to engage the basic symbolic tools implemented in any system to derive the basic equations for the stochastic perturbation technique and how to make an efficient implementation of the semi-analytical methods using an automatic differentiation and integration provided by the computer algebra program itself. The second important illustration is probabilistic extension of the finite element and finite difference methods coded in MAPLE, showing how to solve boundary value problems with random parameters in the environment of symbolic computing. The response function method belongs to the third group, where interference of classical deterministic software with the non-linear fitting numerical techniques available in various symbolic environments is displayed. We recover in this context the probabilistic structural response in engineering systems and show how to solve partial differential equations including Gaussian randomness in their coefficients.

scholarly journals Stochastic Sensitivity Analysis of Cylindrical Shell

2016 ◽  
Vol 16 (2) ◽  
pp. 35-42
Author(s):  
Maksym Grzywiński
Keyword(s):  
Sensitivity Analysis ◽  
Thin Shell ◽  
Taylor Expansion ◽  
Perturbation Technique ◽  
Stochastic Perturbation ◽  
Random Variable ◽  
Engineering Practice ◽  
Stochastic Sensitivity ◽  
Stochastic Sensitivity Analysis ◽  
Stochastic Perturbation Technique

Abstract The paper deals with some chosen aspects of stochastic sensitivity structural analysis and its application in the engineering practice. The main aim of the study is to provide the generalized stochastic perturbation technique based on classical Taylor expansion with a single random variable. The study is illustrated by numerical results concerning an industrial thin shell structure modeled as a 3-D structure.

scholarly journals Stochastic Elasto-Plastic Analysis of Necking Bar with Random Tvergaard Coefficients

2019 ◽  
Vol 252 ◽  
pp. 05002
Author(s):  
Michai Strąkowski ◽  
Marcin Kamiñski
Keyword(s):  
Least Squares Method ◽  
Perturbation Technique ◽  
Computational Modelling ◽  
Stochastic Perturbation ◽  
Random Variables ◽  
Structural Response ◽  
Design Parameters ◽  
Analytical Technique ◽  
Porous Plasticity ◽  
General Order

This paper reports on the computational modelling of static extension tests of the round steel bar. The main objective was to apply the generalised stochastic perturbation technique implemented as the Stochastic Finite Element Method to carry out the numerical simulation of its elasto-plastic behaviour. This approach was based on: the general order Taylor expansion of all input random variables and the resulting state functions of their average means, as well as on the Least Squares Method employed to determine analytical functions of in-between design parameters and the given structural responses. Tvergaard coefficients were assumed as the uncorrelated Gaussian random variables to check the effect of material porosity uncertainty on the statistical scattering of its deformations and stresses. The computational implementation employed the FEM system ABAQUS and computer algebra system MAPLE, including polynomial and non-polynomial local response functions of the displacements, plastic strains and reduced stresses. Moreover, 4-node axisymmetric, continuum, reduced-integration FEM elements (CAX4R) were used in the conducted analyses. The basic probabilistic characteristics of the structural response (expectations, coefficients of variation, skewness and kurtosis) were determined throughout the entire deformation process as the functions of input uncertainty level. The obtained results were finally contrasted with the classical Monte-Carlo Simulation scheme and the semi-analytical technique for input coefficient of variation of porous plasticity coefficients not larger than 0.20.

scholarly journals Vibrations And Deformations Of Moderately Thick Plates In Stochastic Finite Element Method

2015 ◽  
Vol 15 (2) ◽  
Author(s):  
Maksym Grzywiński ◽  
Iwona Pokorska ◽  
Mariusz Poński
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Perturbation Technique ◽  
Stochastic Perturbation ◽  
Random Variable ◽  
Dynamical Analysis ◽  
Thick Plates ◽  
Moderately Thick Plates ◽  
Stochastic Perturbation Technique ◽  
Element Method

Abstract The paper deals with some chosen aspects of stochastic dynamical analysis of moderately thick plates. The discretization of the governing equations is described by the finite element method. The main aim of the study is to provide the generalized stochastic perturbation technique based on classical Taylor expansion with a single random variable.

scholarly journals Probabilistic analysis of random nonlinear oscillators subject to small perturbations via probability density functions: theory and computing

2021 ◽  
Vol 136 (7) ◽  
Author(s):  
Juan-Carlos Cortés ◽  
Elena López-Navarro ◽  
José-Vicente Romero ◽  
María-Dolores Roselló
Keyword(s):  
Steady State ◽  
Probability Density ◽  
Perturbation Technique ◽  
Stochastic Perturbation ◽  
Random Variable ◽  
Spectral Functions ◽  
Power Spectral ◽  
Stochastic Perturbation Technique ◽  
Stochastic Perturbation Method ◽  
Gaussian Stochastic Processes

AbstractWe study a class of single-degree-of-freedom oscillators whose restoring function is affected by small nonlinearities and excited by stationary Gaussian stochastic processes. We obtain, via the stochastic perturbation technique, approximations of the main statistics of the steady state, which is a random variable, including the first moments, and the correlation and power spectral functions. Additionally, we combine this key information with the principle of maximum entropy to construct approximations of the probability density function of the steady state. We include two numerical examples where the advantages and limitations of the stochastic perturbation method are discussed with regard to certain general properties that must be preserved.

scholarly journals Stochastic Finite Element Method for Modelling Heat Transfer in the Building Envelope

2020 ◽  
Vol 172 ◽  
pp. 08007
Author(s):  
Witold Grymin ◽  
Marcin Koniorczyk
Keyword(s):  
Heat Flux ◽  
Monte Carlo ◽  
Energy Demand ◽  
Perturbation Technique ◽  
Carbon Dioxide Emission ◽  
Stochastic Perturbation ◽  
Climatic Conditions ◽  
Differential Method ◽  
Building Envelope ◽  
Stochastic Perturbation Technique

Improving the energy efficiency of the buildings is one of the most effective and fastest ways of reduction of the carbon dioxide emission. However, in the assessment of the energy demand of the buildings, numerous factors are uncertain, i.e. layer thickness, material parameters, climatic conditions, etc. In the present study, mathematical model was developed for analyzing temperature distribution and heat flux in the wall with thermal conductivity of insulation as a random parameter. The results obtained employing a stochastic perturbation technique were compared against the results of the Monte Carlo simulation. Stochastic perturbation technique has been implemented using the tenth order Taylor series expansion. The direct differential method was used to determine the values of Taylor’s coefficient. The obtained results indicate good accordance of the stochastic perturbation technique with the Monte Carlo method. Afterwards, the expected value of the heat flux and its variance were studied for the reference year for a city in the Central Europe. Two cases of the external wall were investigated, in which the thermal insulation was localized either on the internal or the external side of the wall. Performed analyses serve as a good method for assessing the reliability of results obtained using standard, deterministic approach.

Sign in / Sign up

Export Citation Format

Share Document